SYDE372- Lab 1: Clusters and Classification Boundaries Solved

1         Purpose
This lab investigates three related areas: calculating orthonormal transformations, creating decision boundaries, and assessing classification error.

You may use whatever software you like in order to complete the lab, but Matlab is strongly encouraged.

Online Resources:

•    A brief overview of Matlab, as well as a routine for plotting ellipses (plot ellipse.m), are available on the course web page: http://ocho.uwaterloo.ca/⇠pfieguth/Teaching/372/sd372.html • Grammar, report writing, and figure advice:

http://ocho.uwaterloo.ca/⇠pfieguth/Teaching/grammar.html

•    ”Getting Started with Matlab ” tutorial in PDF format: http://www.mathworks.com/access/helpdesk/help/pdf  doc/matlab/getstart.pdf

•    ”Matlab Summary and Tutorial”:

http://www.math.ufl.edu/help/matlab-tutorial/

Class Data
In this lab, consider five classes with the following bivariate (i.e., n = 2) Gaussian distribution parameters:

CASE 1:

                          Class A: NA = 200 µA = [5 10]T                        ⌃ 

                   Class B:      

CASE 2:

                       Class C: NC = 100 µC = [5 10]T                         ⌃ 

Class D:  

                       Class E: NE = 150 µE = [10 5]T                         ⌃ 

2         Generating Clusters
1.   Use the Matlab function randn to assist in the generation of the 2D clusters above.

The randn function will produce normally distributed data with mean 0 and variance 1.0. To create the correlated data as required, you will need to apply a transformation to the uncorrelated, equal-variance data.

2.   Plot the samples and the unit standard deviation contour for each ofthe four classes. Put Classes A and B together on one plot; C, D, and E together on another. Visually, how does the unit contour relate to the cluster data?

3         Classifiers
For the two cases, plot the classification boundaries between the classes using

1.   Minimum Euclidean Distance (MED), using the true means as theprototypes.

2.   Generalized Euclidean Distance (GED), using the true means and covariances.

3.   Maximum A Posterioi (MAP), using the true statistics. Set the a priori class probabilities proportional to the number of samples in each class.

4.   Nearest neighbor (NN), using Euclidean distance.

5.   k-Nearest neighbor (kNN) for k = 5, using Euclidean distance.

For each case, plot the class samples, unit standard deviation contours, and the MED, MICD and MAP boundaries on the same plot, and the class samples with the NN, 5NN boundaries on a separate plots. An analytical expression for the classification boundaries is not required nor ever desired; approach the problem numerically (e.g. create a 2D grid, classify each point, then do a contour plot). Using di↵erent plot line styles (try help plot in Matlab) will make the figure clearer.

Comment on the classification boundaries. How do the di↵erent boundaries compare?

4         Error Analysis
For each of the two cases, determine

1.   The experimental error rate P(✏), and

2.   The confusion matrix for each classifier (MED, MICD, MAP, NN,5NN)

As NN, 5NN are a function of the individual data points, you will need to generate separate training and testing sets.

Compare the results. Which error is smallest? What do you observe in the confusion matrices for CASE2?

5         Report
Include in your report: • A brief introduction.

•    Discussion of your implementations and results (Include brief derivations, as appropriate, for equations implemented in M-files. Don’t bother generating equations using a word processor. Handwritten equations are ok as long as they are readily legible.)

•    Printouts of pertinent graphs.

•    M-files for each section.

•    Include answers to all questions.

•    A brief summary of your results with conclusions.

Keep your report short! We are not looking for length.